LIMIT THEOREMS FOR VARIATIONAL SUMS · tion of all such bivariate limit distributions is obtained....

TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 191,1974

LIMIT THEOREMS FOR VARIATIONAL SUMS

BY

WILLIAM N. HUDSON AND HOWARD G. TUCKER

ABSTRACT. Limit theorems in the sense of a.s. convergence, convergence in

L j-norm and convergence in distribution are proved for variational series. In the

first two cases, if g is a bounded, nonnegative continuous function satisfying an

additional assumption at zero, and if )Xu), 0 < t < T\ is a stochastically contin-

uous stochastic process with independent increments, with no Gaussian compon-

ent and whose trend term is of bounded variation, then the sequence of variational

sums of the form 2? g(X(t •) — X(t , ,)) is shown to converge with probability

one and in L ,-norm. Also, under the basic assumption that the distribution of the

centered sum of independent random variables from an infinitesimal system con-

verges to a (necessarily) infinitely divisible limit distribution, necessary and

sufficient conditions are obtained for the joint distribution of the appropriately

centered sums of the positive parts and of the negative parts of these random

variables to converge to a bivariate infinitely divisible distribution. A charac-

terization of all such limit distributions is obtained. An application is made of

this result, using the first theorem, to stochastic processes with (not necessarily

stationary) independent increments and with a Gaussian component.

1. Introduction and summary. The expression "variational sum" refers

to the sum of a function of independent random variables from an infinitesimal

system of random variables. More precisely, let <'X ,,•••, X , ! i be an infinites-

imal system of random variables, i.e., for 72 = 1, 2, • • •, the k random variables

Xnl,"., Xnjt ate independent, and, in addition, k —* 00 as 72 —»00 and for every

sequence of integers i/n! such that 1 <j <kn,Xn. —* 0 in probability. Then,

for a real-valued or vector-valued function g, the expression 2.". g(X .) is a

variational or g-variational sum. Three interrelated limit theorems on variational

sums are proved in this paper.

The first of these theorems, Theorem 1 in §2, yields a conclusion of almost

sure convergence and L j-convergence for variational sums of the form

2?_j g(xh„¡> - xh„ ,•_,.))• Here |X(z), 0 < t < T] is a stochastically continuous

stochastic process with independent increments, no Gaussian component, and a

trend function of bounded variation over [0, T], The function g is a bounded,

Presented to the Society, January 26, 1973 under the title Bivariate limiting distribu-

tions for variational sums; received by the editors November 27, 1972 and, in revised form,

March 8, 1973.

AMS (MOS) subject classifications (1970). Primary 60F05, 60G99.

Key words and phrases. Variational sums, infinicely divisible distributions,

stochastic processes with independent increments.

Copyright © 1974, Americio Míthenuticil Society

405

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

406 W. N. HUDSON AND H. G. TUCKER

nonnegative, continuous function over R which satisfies g(x) <M|x|^, where

max il, ß\<y<2 or f3 = y=2 and ß = inf \8 > 0: Jjx|sl |x| sMr(ax) < «}, Mix)

denoting the Levy spectral measure corresponding to X(i). This result extends

theorems of S. M. Berman (Theorem 5.1 in [l]), R. Cogburn and H. G. Tucker

(Theorem 2 in [3]) and P. W. Millar (Theorem 4.5 in [7]).

The second such theorem deals with the joint limiting distribution of the sum

of the positive patts and the sum of the negative parts of random variables from an

infinitesimal system. In particular, let ÍÍXnl, • • •, X , }} be an infinitesimal sys-

tem of random variables as defined above. The general hypothesis is that there

exists a sequence of real numbers ¡ani such that the distribution function of

2 ■ "j X. + a converges to some limit distribution at all its points of continuity.

This limit distribution is necessarily infinitely divisible and has a characteristic

function of the form

/(a) - expjya - <¿£ +£ (e*>* - 1 - ff^***\

where y and a > 0 are constants and Mix) is a nondecreasing function over

(-oo, 0) and over (0, oo) such that M(-«) = M(+oo) = 0 and j-]_ x x2 dMix) < oo. For

X any random variable let us denote X = Xkx¿Q-\ and X = -X/r s0"|, so that

X = X - X _ xhe problem considered is: given the above general hypothesis,

under what conditions do there exist sequences of constants iaßi and \v { such

that the joint distribution of the pair of centered sums

y>+. + a , Y. X'. + vmf-' nj n *-^ nj n

7=1 7=1

converges as n —► oo to that of a bivariate infinitely divisible distribution func-

tion? Here g: R' —» R is a function defined by g(x) = (x+» x~). In §3 necessary

and sufficient conditions are obtained for such a limit law to exist. This theorem

is an improvement of Theorem 1 in [8] and a result of Loève [5]. A characteriza-

tion of all such bivariate limit distributions is obtained.

In §4 there is a third such theorem. It is an application of the first two

theorems to separable stochastic processes with independent increments which

are centered and have no fixed discontinuities (i.e., Xit — O) = X(/ + 0) a.s. for

each t). In particular, if }X(r), 0 < t < T\ is such a process, then letting 0 =

t 0< t x< • • • <t =Tbea sequence of partitions under refinement such that

maxi/. . — / . ,; 1 < j < n\ —► 0 as n —* oo, one considers for each such parti-nj n,]—i — ' — *

tion the sum of the positive parts and the sum of the negative parts of the incre-

ments. Under suitable centering the bivariate limit distribution is shown always

to exist. The results of §4 might overlap with those of P. Greenwood and B.

Fristedt [4] who treat the more special case of processes with stationary inde-

pendent increments.


LIMIT THEOREMS FOR VARIATIONAL SUMS 407

2. Extension of a theorem of Berman. In this section a limit theorem is

proved for g-variational sums for stochastic processes with independent incre-

ments. The limit here will exist both in the almost sure sense and in L j-norm.

The stochastic process is one with (not necessarily stationary) independent incre-

ments, and the function g is from a rather general class. Its interest is that for

the most part it extends a theorem of S. M. Berman, namely, Theorem 5.1 in [l]. Also

extended in a sense are Theorem 2 in [3] and a theorem of P. W. Millar, Theorem

4.5 in [7]. An application of this theorem will be given in §4.

In this section and in §4, \Xit), 0 < t < T] will be a stochastic process with

independent increments which is centered, has no fixed points of discontinuity

and is separable. The characteristic function of Xit) can be written as

fXU)iu) = exp{za(i)u - £ÎkV + ¡-0 + f~ 7>* _ , _ ff-,)Mtidx)),

where a(i) is a continuous function over [O, T], o it) is a continuous nonnega-

tive, nondecreasing function, and Mi ) is the Levy spectral measure whose

properties are given in a number of places, e.g., Loève [6]. 7tz addition, it is

assumed that ait) is of bounded variation over [O, T]. Such a process Xit) can

be written as a sum of two independent separable stochastic processes Uit) and

V(/). The process Wit), 0 < t < T], called the Gaussian component, has charac-

teristic function /fj(7-\(a) = expi-a it)u ¡2], and it is known that almost all of

its sample functions are continuous over [O, T], The non-Gaussian component

has characteristic function

fVU)iu) = exp{zaG)> +£ +£ f>* - , - ^)«t<*4

and almost all of its sample functions are right continuous and have left limits at

each point. Each such sample function necessarily has discontinuities of the

first kind only, and a countable number of them at that. Now, for every positive

integer 72, let 0 = r . < r .<•••</ =Tbe such that0 77,U 77,1 77,72

!/770',,,'ínn!Clín + l,0',"'/n + l,77 + l!

and such that max if , - t , ,: 1 < k < n] —» 0 as « —»00. We shall use the77, K. 71,K— 1 — —

following notation in these two sections: Xnk = X(r ,) - Xitn ¿_j)» F , denotes

the distribution function of X ,,

%k = <*„*> - "<V*_x>> unk = uitj - uitn< k_ j), vnk = vitnk) - vítn¡k_ j),

and

ß = inf /S >0:j\ \x\SM Jdx)< ool.



The number ß is an index originally due to Blumenthal and Getoor [2] and gen-

eralized by Berman [l]. For every s e (0, r], we define Jis) = X(s) - Xis - 0) =

Vis) - Vis - 0) as the jump at s. An arbitrary point in "the probability space is

denoted by <u. The theorem to be ptoved in this section follows.

Theorem 1. Let g be a bounded, continuous, nonnegative function defined

over R1. Suppose there are constants M > 0 and y such that ye(max(l, ß), 2]

or ß = y=2 and g(x) < M|x|r for all x. Let \Xit), 0 < t < T\ be the stochastic

process defined above but without a Gaussian component. Then

(A) Z fgMdFnkix) -+f gix)MTidx)fc=l

as n —► oo, and

n

(B) E^XJ-£<g(/(*)): 0<S<T}k=l

almost surely and in Lomean.

It should be noted that if we required g(x) < K\x\y to hold in a neighborhood

of 0, then by the boundedness of g there exists an M > 0 such that g(x) < M |x | ̂

for all x.

By the size of a jump of the process at s we mean |/U)|. Let us denote

/ ■ (i) as the random variable which is 1 if no jumps of size greater than e occur

in it i_j, t A and which is zero otherwise. We also need the notation

Y«\t) = Xit) - ait) - £ ,/(*): \]is)\ > e, 0 < s < t\

and

*S » *nk -*nk-Z ^S)-- l/WI >^tn,h-l<S< 'n**'

Lemma 1. // max i 1, ß\ < y < 2, then

C ilim lim sup ElYgíX ,)l .íe )} = 0.

m^oo n^oF J^6 "k n, m j

Proof. Since g is nonnegative, and since X. = Y. + a whenever

/ (i ) = 1, we have, for every n and ;',nj m ' ' *

giX )I le ) < gÍY^f + a ).5 n; nj m —° nj nj

Hence

Now by hypothesis g(x) <M|x|r, so



n

(2) BjL log-'+ an?\ i ME\Z K? + aJ

Since y > 1, we may apply the inequality \a + 6|y < 2y~ i\a\y + \b\y) (see

Loève [6, p. 155] to obtain

(3) EJilrS»' + a„.rj < 2^"1EJ¿|yM|r| + **-« £ |aj*.

Now77

(4) limsup^|an.|r=077-.00 y_ J

since <x(i) is continuous and of bounded variation over [0, T] and y > 1. We

obtain

(5) un, limsupE ¿.|y^mV =0m-wo „_oo ^._j |

by applying two lemmas due to S. M. Berman, namely, first applying Lemma 5.3

and then applying Lemma 5.1 in his paper [l]. Combining (1)—(5), we obtain the

lemma. Q.E.D.

Lemma 2. For every m,

Um E ¿ g(X J(l - Ijcjl = e(£ {gijis)): \Jis)\ ><m,0<s< rl)

= /i i gix)MTidx).I*l>f771

Proof. Let X = supig(x): x £ R1! < oo. Let N denote the number of jumps

during [0, T] of'size greater than cm. Then N is a random variable; it has the

Poisson distribution with expectation /. , Mjdx)< oo. Thus, if we denote

Z = 1" giX . )(l - 7 Ac )), we have 0 < Z < X,V a.s. In addition, Z -,77. 76=1 ° 77« 77K777 — 77— '77

Sjg(/(s)): \Jis)\ > e , 0 <s< T] a.s. because almost all sample functions are

right continuous and have left limits at every point and a finite number of jumps

of size greater than f , and because g is everywhere continuous. Thus, the first

equality is obtained by the dominated convergence theorem. It is well known that

the last two terms in the statement of the lemma are equal. Q.E.D.



Lemma 3. The following holds:

lim £gíXnk) = r,\gíJís)):0<s<T\a.s.

Proof. By hypothesis on M_ and since ß < y < 2 or /S = y = 2,

2\\]is)\y: 0<s <T, \]is)\ <li has finite expectation £x |x|rMT(¿x), and

hence it is a finite random variable. Thus, for arbitrary e > 0 and r¡ > 0 there

exists a k > 0 (k < l) small enough so that

(D P[Z t l/M y- 0<s<T, \]is)\ < k! > e/ÍM + 1)] < t,,

whete the constant M is from the hypothesis on g, and such that ±zc are con-

tinuity points of MT(x). Let us denote An = ^_lg0(nk)lnkÍK) and Bn =

2£=1g(Xnfc)(l - '„¿(x)). Let {/(i), 0 < t < T\ be a stochastic process defined by

y(i) = X(r) - X {/(*): 0 < s < i, |/(5)| > k¡,

and denote Y , = Y(i ,) - Yit , A. We now show thatnfc tz« n,lt— 1

n

(2) A„ - Z ¿?(ynifc) — ° a-s- as tz -. 00.

Indeed, since X(«) is continuous a.s. at each fixed / e [O, T], there exists an

event ÖQ of probability one such that, for each fixed tu eiiQ, X(r, cu) is con-

tinuous at all the tnk's. Take <y e QQ. For each tz there are no jumps at any of

the endpoints of any of the subintervals. For each tz there are at most Nía) sub-

intervals which are not included in the sum defining A but which are included

in the sum ^l-iB^nk)' w^ere ^ is a Poisson random variable with expectation

/. j M-idx) < ». Ih each of these Nicj) subintervals the point at which a jump

of size > k for X(r) was located is a point of continuity of Yit). Hence, con-

tinuity of g at zero implies each of these NÍoj) summands tends to zero as

n —» 00 , which proves (2). In the proof of Lemma 2, it was established that

(3) Bn— Y,\gijis)):0<s<T, \]is)\ > k\ a.s. as»-».

Thus, by Theorem 5.1 in Berman [l] in the case max il, ß\ < y < 2 or by

Theorem 2 in [3] in the case ß = y = 2, we have by hypothesis on g that

n n

lim sup X giYj < M lim sup £ \Yjy(4) n-°° fe=i n-00 *=i

= M£i|;(s)|r:0<s<T, |/(s)|</<!.

Hence, applying in order (3), (2), (4) and (1) we obtain


LIMIT THEOREMS FOR VARIATIONAL SUMS

r I " 11' lim sup £ gíXnk) - Y,{gíjís)): 0 < s < Ti > eL n-°° U=l J

411

<P 1lim sup71-»OO

+ lim sup77-.00

A„ -Hk(Jís)): 0<s<T, \jis)\ < k]\

B„ - Z lg(/(«)): 0 < s < T, \jis)\ > k]\ > f

= P lim sup An-^jg(/(s)):0<s<r, |/(s)|<#cl >eL n-"°° I J

= P lim sup ¿ gÍYnk) - £íg(/U)): 0 < s < T, \jís)\ <k] > f

L "-00 k=l I J

< P[(l + M) £i 1/(5)^: 0 <s < T, \ris)\ <«!>*]< r),

which proves Lemma 3 by the arbitrariness of « and n. Q.E.D.

Lemma 4. // ß = y = 2, í&erj fee sequence ¡2" g(X .)} z's uniformly inte-

grable.

Proof. Let Yit) and Y . be as defined in the proof of Lemma 3, and let

Yn = 2" y2.. h is known (see, e.g., S. J. Wolfe [ll, Theorem 2]) that each Ynj

has moments of all orders. We first compute EYn. Let us define

M . = M -M. and a . = ait .) - a(/„ , ,).

The characteristic function of Y . is"7

/y .(n;

a) = exp/zX . - f -*— M.idx)) + f (V"* - 1 - JíoUm Í&A.^\ \n, J\x\zKl+x2 «J ) J\x\<k\ l+xy n, j

Differentiating twice with respect to a, and recalling that FY . = — fy ,(0) and

MT = 2? M we obtain

Ey = Y EÍY2)n t-^ n,

7=1

(2)

~\ "> J**M+*2 "' J|x|<:k i +x2 », J7=1 x

+ T x2M_(ax).J\x\<K '

Let y = l{ÍJÍs))2: 0 <s < T, \jis)\ < k\. Then it is known (see, e.g., Lemma 5

in §3 of [2]) that Ey = f, , K x2MTidx). We now wish to prove that Eyn —» Ey.



To do this, it is sufficient to prove that Tn —» 0 as tj —» oo, where T is defined by

,2r =Tíhít )-bit . .))ni-* tz; t2,;-1

7 = 1

and

hit) = ait) - f -S-, M ,(ax) + f , -ii, M (¿x).

We first make two observations:

(i) Since M(ix) is continuous in t tot each x at which M^XO is continuous,

we have by the Helly-Bray theorem that f<x\iK U/(l + x2))MA\dx) is a continuous

function in /. Since M as a measure is nondecreasing with increasing t,

f K (x/(l +x2))Mtidx) and -fXSm,K (*/(l + x2))M Jjx) ate nondecreasing and

continuous as functions of i.

(ii) Consider the finite measure pnÍA) = fA x2M .(ax) defined over all Borel

subsets A of [— k, k]. For 1 < / < tz, X . —► 0 in probability, and hence by a— TZ "In

known theorem on convergence of sequences of infinitely divisible distributions

(see, e.g., Theorem 2 on p. 163 of [4]), M . (x) —» 0 as n —» oo at all x /= 0. Thus,7fj

ip ! converges to the zero-measure over [-K, #c], and hence by the Helly-Bray

theorem,

maxif , -J¿_ M idx): 1< / < rzl — 0Vl*l<« 1+X2 "' ~ )

as tj —> oo. This implies that, as a function of t, fixi<K(x Al + x2))M idx) is

continuous, and, in addition, /0<x<KU3Al +x2))M{dx) and-/_K<x<0(x3/(l +x2))MÍ4x)

ate nondecreasing and continuous functions of i. Observations (i) and (ii) and the

hypothesis on a(.) imply that hit) is continuous and hit)- ait) is equal to a

difference of two nondecreasing continuous functions, and hence h is both con-

tinuous and of bounded variation over [0, T]. We may write

T < max \\hitn)-hítn>._x)\\ ¿ \hítn)-hítn._x)\.

The conclusion drawn from the two observations implies Tn —» 0 as n —» oo.

Thus we have shown EY —» EY as tz —» oo. Now, by Theorem 1 in [3], Y —, Yn ' * ' n

a.s. as tz —»00. Since \Y } ate nonnegative, these two results are easily seen to

imply E|Y - YI —> 0 as tz —> 00, which in turn implies that \Yn\ is uniformly

integrable. Let us denote S =?."_ giX .)/ .(«). Since by hypothesis gix) < Mx2

tot all x we have

S <TgiY )<MYY2.,n-*-*6 n] — i-i nj '

7=1 7=1

and by what we have just proved, we obtain that ÍS ! is uniformly integrable. We



obtain the conclusion of the lemma from the inequality

7 = 1

We now prove (A) and (B). By Lemma 3 we immediately obtain the almost

sure part of (B). This much of the conclusion and Lemma 4 imply in the case

y = j8 = 2 conclusion (A) and the L j-mean convergence part of (B). We now prove

conclusion (A) of Theorem 1 in the case max{l, ß] < y < 2. Indeed, one easily

sees that, for every m,

lim sup e(¿ gíXjíl - lnjA < lim sup e(£ gixÀn-00 \*=1 ' "-00 U = l /

< lim sup e/¿ giXjil - lnjcjj+ lim sup e(¿ giXjljA

These inequalities plus Lemmas 1 and 2 imply

lim sup e( ¿ g(X J = Jim lim sup e(¿ giXjíl - Ijcj)]

= lim J gix)M idx)=fgix)Mjdx),m_oo |x|>em

which proves (A) in the case max ¡1, ß] < y < 2.

Now we use a well-known result which states that if Z and Z are nonnega-

tive random variables with finite expectations, if Z —» Z a.s. and if EZ —» EZr 77 77

as 72 —» oot then E|Z - Z| —»0. Thus, by conclusion (A) and Lemma 3 we obtain

conclusion (B) in the case maxll, j8! <y < 2. Q.E.D.

3. Existence and extent of bivariate limit laws. The univariate and bivariate

forms of the general central limit theorem are used repeatedly for the rest of this

paper. The general multivariate form is given here for easy reference. (See

Theorem 2.3 in [8, pp. 190-191]).

Central limit theorem. Let {{X ,,•••, X , ]] be an infinitesimal system of

p-dimensional random vectors, and let H . denote the distribution function of X ..r 'n, ' ' 77,

7t2 order that there exist a sequence of constant p-dimensional vectors jc ! such

that S."j X . + c converges in law it is necessary and sufficient that there

exist a completely additive, nonnegative set function NÍS) defined on Borel sets

not containing 0 and a nonnegative quadratic form Q(t) defined over t e R^ such

that

(i) For Borel sets S which are continuity sets of N and which are bounded

away from zero,



*" C

X I, dH .(x) -» NÍS) as a — oo,

7=1

(ii) limïïm'vjf, it'x)2dH íx)-(( t'xdH .(x)VLß(t).

The p-dimensional vectors je j may be chosen by the formula

c„ = ¿Z f ! *dF (x)-y,» *^Jx<r »; '

7=1 '"

where r> 0 is arbitrary and y is a constant vector. The logarithm of the

characteristic function of the limit distribution is given by

log /(t) = iy't-Q{i)/2 + f , [exp(zt'x) - 1 - (t'xKl + |x|2)"'] Nidx).x>0

Now let i{X .|| be an infinitesimal system of random variables (as defined

in §1), let F . be the distribution function of X ., and recall that the general

hypothesis means that there exists a sequence of real numbers fa j such that the

sequence of centered sums S.™ X . +a converges in law. For every positive

integer n and every e > 0, let us define

o2ie, n, +) = Z {J/x2dFn.íx)-(f;xdFn}ix¡\2},

o2ie, n, -) - £ {Zx2dFn.ix) - (£ xrfF./x))},

and

Cr(N-)-£{jT«*'«/»)iCf«aFjU)}.;'=l *

Theorem 2. Under the general hypothesis given above, necessary and

sufficient conditions that there exist sequences of real numbers ¡a ! aT2¿ \v \

such that the joint distribution function of X". X . + u and S.n, X . + v' ' ' ; = 1 ti; n ; = 1 n; n

converges to a limit distribution are that there exist constants ax x > 0 and

a22 > 0 such that

(D ifti^(wP)ff2U'B*+)-ff»

aTzrf

O) lim lim (. . )o-2(í> n, -) = ff„.W fi0 Tt-.oo\inf/ 22



Proof. Under the general hypothesis, {{Xnl,- ■ •, Xnk ]] and HXBl,«",Xnfc ]]

ate infinitesimal systems. If there exist sequences of real numbers {uj and {vj

such that £.", X . + u and 2." X . + v converge jointly in law, then the, = 1 71, 77 , = 1 77, 77 S 1 J >

centered sums each converge in law, and the central limit theorem shows that (1)

and (2) are necessary. The proof of the sufficiency of (1) and (2) is similar to the

proof of Theorem 1 in [8] and is given here for the sake of completeness. Define

S = {(xj,x2)eR2: Xj = 0, x2 >0 or Xj > 0, x2 = 0i.

Let

Gn;.(xj,x2) = P[Xn.<Xj,Xn".<x2].

Then for any Borel set B C R ,

fB dGn.ixj,x2) = 0 if BO S =0,

= I a"E .(x) otherwise,JCB 77,

where Cfi = {x > 0: (x, 0) £ B ] u {x < 0: (0, -x) e B ]. By hypothesis and the

central limit theorem, we have

lim T ÍF lx) - 1) = Mix) it x > 0,

<3> *

lim £ F .(x) = M(x) if x < 0,"^ ,=1

where in both cases x is a continuity point of M. Hence there is a Levy spectral

measure N over R2\j(0, 0)! such that

Í4) lim X \dGn.ix.,x,) = /V(B)= f a-M(x)77-.00 • . ö J •'Cß

for every Borel set ß C R whose closure does not contain (0, 0) and such that

7V(<?B) = 0. Let us define

7 = 1

One easily verifies that

ff2(e, 72) = e2ie, n, +) + a2íe, n,-)-2 Críe, n).

By the general hypothesis and the central limit theorem we know that

(5) lim lim /S"Pl(72(f,72) = a2.ej0 Ti-.» (ml }

Now, (1), (2) and (5) imply



(6) lim lim <SUf>Crie, n) = o\,,fiO rz-oo (inf j I2

for some nonpositive constant 0"12 which satisfies o2 =axx + a - 2a,,. Let

t «» Uji £2)» x = (*i» x2) and

ß.W = I {J¡x|<f (t. ̂ *^W - (JM<f (t. x)z/Gn;.(x))2},

where (t, x) ■ t^x. + t2x2. One observes that

Qnil) = a2ie, n, +)t\ + o2ie, n, -)r2 + 2txt2 Crie, n).

Applying (1), (2) and (6) to this expression, and taking (4) into account, we ob-

tain the sufficiency of (1) and (2) by the multivariate form of the central limit

theorem given above. Q.E.D.

It should be pointed out here that it is possible that the general hypothesis

above holds and yet conditions (1) and (2) are not both fulfilled. An example of

such can be easily deduced. Indeed, in [9] a distribution function Fx of a ran-

dom variable X was constructed so that neither Fx+ nor F is in the domain

of attraction of the normal distribution and yet Fx is. Thus, if {X } is a se-

quence of independent identically^ distributed random variables with common dis-

tribution function Fx and with normalizing coefficients }B }, and if we denote

Xn£ = Xj/Bn, 1 < k < n = k , then it is easy to see that the general hypothesis

is satisfied but that neither of the conditions of Theotem 2 is satisfied.

We now present the set of all bivariate limit distributions obtained in

Theorem 2.

Theorem 3. Under the conditions of Theorem 2, the joint limiting bivariate

distribution obtained bas a characteristic function of the form

(1) fiux, uA = exp/zy'u - u'Su/2 +J"~ Aíux, x)Mxidx) +J"~ A(a2> x)M2(ax)\,

where S = (a..) with oX2=a2X<0, Aiu, x) = eiux - 1 - z'ax/(l + x2), M j(x) =

M(x), and M Ax) = -M(-x), where M is the Levy spectral measure determined

under the general hypothesis. Conversely, given any Levy spectral measure M

and a bivariate distribution of the above form (1) with this M, there always exists

an infinitesimal system ÍÍX ., 1 < / <*„!! which satisfies the general hypothesis

and the conditions of Theorem 1 so that the limiting distribution of appropriately

centered sums of the positive and negative parts is given by (1).

Proof. We need only prove the converse, the direct statement following

immediately as a corollary to Theorem 1, using the multivariate form of the central

limit theorem quoted earlier and the fact that Críe, n) < 0.



In order to prove the converse, we show that if y = (y15 y2) e R , if ¿ = ía^)

is any two-by-two nonnegative definite symmetric matrix for which CTj 2 = a21 < 0,

and if M is any Levy spectral function, then there exists an infinitesimal system

UX , 1 < / < k ]] which satisfies the general hypothesis and the conditions of

Theorem 1 so that the limiting distribution of properly centered sums of the posi-

tive and negative components is of the form given in (1).

First we shall show that, given any matrix 2 just described, there is a se-

quence of independent, identically distributed random variables |X { such that,

for some sequences of normalizing constants B —» +00 and centering constants

{aj and {ßj, then the joint limiting distribution of B~12B=1Xn + a^,

B~ £" , X. + ß is bivariate normal with zero mean vector and covariance77 , = 1 , r~n

matrix £. We must consider four cases.

Case (i). (7j2 = <72 j < 0, det S > 0.

Consider a random variable X which takes values -a, 0, b with probabilities

(1 - 0)/2, 0, (l - d)/2 respectively, where

6 = ((o"na22^ + CT12^2^°llff22 " a12^'

b = 2aft/(l - 62ÍÁ, and a = 2ff*/(l - d2YA.

One can easily verify in this case that 0 < 0 < 1, var(X ) = a.., var(X ) =

a22 and EX EX" = -o\,. Now let [X | be independent identically distributed

random variables with the same distribution as X. By Theorem 4 in [9] we may

take B = »/*. a =-rz^EX and ß =-n'2EX~ and obtain the conclusion77 T 77. ^ 77

sought in this case.

Case (ii). a., = a,, < 0, det S = 0.

Consider a random variable X which takes values 2ff j j and -o"^ w'tn

equal probability M. Apply Theorem 4 from [9] as above.

Case (iii). <7j2 =ff21 = 0, ffjj > 0, a22 > 0.

Let E be a symmetric distribution function, and assume that F is in the

domain of attraction of the normal distribution but not in its domain of normal

attraction. Then it is known that

J**xí2a'F(í)=2joXí2a'F(í)

is a slowly varying function at 00. By Theorem 5 in [9] there exist normalizing

coefficients B —» 00 and centering constants {c ] so that the joint distribution

function of

*;^¿x?+cV B-Jpx-.c^j



converges to that of two independent normal random variables with mean zero and

variance one. Now replace F by

Gix) = Fixoxx*) ifx>0,

= Fixa~22A ) if x < 0.

Let \U ! be independent, identically distributed random variables with common

distribution G, let a - c a,,/B and v = c o~jB . Then the limit distribu-' n n II' n n n 22 n

tion function of

7» n

b~1 y;(/+ + a, b-'t u~ + vn *—> ; n n *-* n n

7=1 7=1

is bivariate normal with mean vector zero and covariance matrix S.

Case (iv). ff12 = a21 = 0; ffjj > 0, ff22 = 0 or ff22 > 0 and axx =0.

In this case, replace G in Case (iii) by

G Ax) = G(x) if x > 0, G (x) = 1 if x > 0,or

= 0 if x < 0, = Gix) if x < 0,

to obtain an analogous result.

In whichever case 2 is above, let F denote the distribution function for

that case, let }B ¡ denote the normalizing constants for F, and denote F.ijc) =

FÍB x), 1 </' <n. Now let € > 0 be such that ±e ate points of continuity of M

and e i 0. For each tz let K = M(-f ) - M(f ). Let \r ! be a sequence of posi-« TJ TZ TZ II * ■

tive integers satisfying Kn/r„ < l/n. If M = 0, take rn = 0. If rn > 0, define

G (x) = M(x)/r , if x<-e.n ft — n

= M(-fB)/rn, if-fn<x<0,

= 1 + Mien)/rn, if 0 < x < fn,

= 1 + Mix)/r , if x > e .n n

Next, define k =n +r and F .(x) = G (x) for n + 1 < ;' < k . Finally, forn tz fj; « — ' — n ' *

every tz let {X ,,.. •, X , { be independent random variables such that the

distribution function of X . is F .. It is easy to verify that ÍÍX .|| is ann; tz; y ' nj

infinitesimal system which satisfies the general hypothesis and Theorem 1. Q.E.D.

4. An application of Theorems 1 and 2. Let ÍX(f), 0 < t < T\ be a stochastic

process which is centered and has no fixed points of discontinuity. The same

assumptions and notation from the beginning of §3 carry over here except that in

this section [Xit), 0 < t < T\ is allowed to have a Gaussian component. The

following theorem is an application of Theorem 1 to X(r), and this section is

devoted to its proof.



Theorem 4. Under the above hypotheses on {XÍt), 0 < / < T]f there is a se-

quence {cj of constants such that the joint distribution ¡unction of S.'jX . + cB

and S." X~.+c cotzfeTges to a bivariate infinitely divisible distribution function

whose characteristic function is

fiuy, u2) = expjzty, u) - u'Su/2 +J"~A(«j, x)dMlix) +J"°°A(a2, xWM 2(x)L

(77-l)a2(D/277 -a2(D/277)277\

277/

(2)

A(a,*) = e'»*-l-zW(l+*2),72(D/277 (77-l)a2(r)/271

and M Ax) = Mjx) and MAx) = -Mj-x), where x > 0 aTza" ±x are continuity

points of Mj,

If X is a random variable with finite expectation, and if A is an event, we

denote E{A, X] = fA XdP. According to Theorem 2 we need to prove that

(1) "fi M ¿ \F{0 < Xnk < e, X2nk] - E2[0 < Xnk < t, Xj] , <* - 1^),

and

lim ¡hj, ¿ÍE[-f<X , <0, X2.]-E2[-e<X . < 0, X . ]} = ^ ~ \)g2(T) ■eio»-00^ nk~ "* "*~ nk 277

Equations (1) and (2) ensure the existence of two sequences, {cj and {c^l, such

that S¿=i *nk + C7j an<* ^fc-i ^ k + cn converge jointly in law. However, since

XÍT) + icn - c') = 2" , X+~+ c - il"t . X" + c'), then \c - c'J is a con-n n K-l 77* n *=1 nc ti fi it

vergent sequence, and hence we may take c = c'.

Equation (1) will be proved for any process satisfying the above conditions.

Now if X(/) is such a process, so is -X(<). Equation (2) follows from equation

(1) by applying (1) to -X(/). Thus it suffices to prove (1) in order to prove

Theorem 4. The first part of the proof is devoted to separating the contributions

of the Gaussian and non-Gaussian components to (1) and showing that the quan-

tity in (1) is due to the Gaussian part alone. An easy computation will finish the

proof.

Lemma 1. The following holds:n

lim lim sup £ E{0 < Xnk < e, V2fe] = 0.ejo 7I-.00 k=l

Proof. It is sufficient to prove for every e > 0 that

77

(*) lim lim sup £ E[0 < Xnk < e, \VnJ < 2e, V2J = 0fiO 7I-.00 ¿_J



and

n

(**) lim £ £[° < Xnk < f» I O > 2e' ̂ J = °-k=l

We first observe that for every tz and k, 1 < k < n,

£[° < xnk < * I^J < 2i> ̂ < «IVJ < *• *^'

An easy consequence of Theorem 1 (where we take g(x) = min íx , 4e \) and the

central limit theorem is that if i 2e ate continuity points of M T, then

«2. Z E^ J < 2e. ̂ =/_2f2£ *2Mr(¿x).&= i

We obtain (*) by taking the limit as e 1 0. We now prove (**). First we recall the

well-known inequality: for A > 0,

1 f" e-*2/2o-2^ < _£_ e-X2/2a2.

V2770-2 JX V2?7

Since U k is normally distributed with mean zero and variance a2^, we have

P[|y„fcl > U] < Í2ank/je^)e-i2(2f'2^.

The monotone convergence theorem implies

ZF[0<Xnk<e,\Vnk\>2e,V2nk]

= Z Z H[0 < Xnk < e, (; + l)e < \V J < (,' + 2)e, V2,].¿fc=l; = l

But Xnk = Unk + Vnjfe; hence, if 0 < Xnk < e and (/ + l)f < \Vnk\ < (/ + 2)e, then

1^ J > /*• Hence the right side above is not greater than

Z Z £t|i/„J > je, íj + l)e < |V J < (/ + 2)e, V2fe].*=1 ; = 1

Because f , and V , ate independent, this is dominated by

Z £(; + 2)VP[|(/J>/f]P[|Vj>2e].¿t = l ; = 1

The upper bound for P[|l/ J > je] determined above implies

£ (/ + 2)VP[|(/J > je] < 3o\k T, ^ e-Ue)2/2"«»;=l '='V2^

< 3(a„,)2/2,



These inequalities imply

77

EF{0<Xnk<£,\Vj>2e,V2k]hml

< Í3a2ÍT)/2) max{o2nJ>{\VnJ > 2t]: 1 < k < n].

Taking the limit of both sides as tz —» oo, we obtain (**). Q.E.D.

Lemma 2. Let A A B denote the symmetric difference between sets A and B.

Then

n

(i) Hm n¡r 2i ßtf° ̂ x„k < «iA to < u„k < & unk) - °-£Í0 »-,0° jt=l

Proof. We observe that

[0 < Xnk < e] A [0 < Unk < e] = A<*' u A<*) U A<3*>,

where

A<*> = [0 < Xn, < c, Eni < 0] u [0 < (/„, < «, Xn, < 0],

A<2*>-[0<XwA<e, €/ÄJb>d and A<*> - [0 < t/^ < «, X^ > <].

Equation (1) above will be proved by showing that

(2) i™ Hm ¿ E[Af\ U2]=0, tot i = 1, 2, 3.

When A(j occurs, X^ and Unk have opposite signs. Since Xnk = Unk +

Vnk, we have A™ C [|E„,| < \Vj]. But i\Uj < \Vj] = A*> Unf» where

AiV - 11^*1 < 1^1 < 2e]' and MS = [\Unk\ < Kß n tl^J >2e3. Now,using the fact that U , and V . are independent, applying Theorem 1 as in the proof of

Lemma 1, and by the Central Limit Theorem, we have

g E ¿EW '•"• »y * ¿ S E ¿ E[A">- "U* K=l J= 1 c 4- u R = 1

« n

<lim lim ^ß[|^l<2f,V2fe] + lim lim £ E[|V J > 2«, E2,]f|0 77-00 fc=1 e|o«-t»fc=i

< ïim P x2MT(<&) + o"2^) ïïm Dm max }P[|V . I > 2e]] = 0,— il— e * i . » ' ufe1 -

£Í0 ' - fiO 77-°° ISftSTZ

which proves (2) for i = 1.

In order to prove (2) when i = 2, we write A2k) = A^ U A 2*' where



A (2*> = [0 < Xnk < e, e < Unk < 2e] and A^¡ = [O < Xnfc < e, Unk > 2e]. Then

lim lim ¿Et^^.tV^ ]<¿lim lim ¿ E[A^, U2 ]flo»-~4=1 y=iflo»-~*=i

<lim lim 4e2 £ PUJfi*) + lim lim Z EH"H*I > <» ̂J'

By independence of U , and V ., the second term to the right of the last ine-

quality is bounded by

ÏÏm ÏÏm" maxlP[|V„J > <]: l<jfe<«lff2(T)eio«-" " "

which is zero. As for the first term on the right of the last inequality, it is

clearly bounded above by

n

ÏÏm" llm" 4í2 Y P[U. > t],clo«-~ ¿7,

which is zero by applying the central limit theorem to the infinitesimal system

\\Un.\\, keeping in mind that 1/(7) = 2J=1 Unj for all tz, and that UÍt) is

Gaussian. This proves (2) for i = 2.

The argument fot i = 3 is similar to that for A2*'. We decompose A(*' as

follows:

A<*>- A(fe)UA<fe)A3 31 u/l32'

where

A<,«-[0<If,a<«/2,Xliik>«I and Aj»)-U/2<l/.à<€.X||4>d.

Since My C [V . > e/2], we have for reasons given before in similar situations,

n n

ÏÏm" ÏÏm" Y E[A\\\ Ü2J <ÏSS ÏÏm" Z E^„ik > ̂ "b*1

< ÏÏm" ÏÏm max }P[V , > e/2]: l<k<n \o2ÍT) = 0.fiO B-OO

Also, since A(*> C [e/2 < Unk < e], we have for by now obvious reasons



77 77

lim lim £ fi[A<*>, U\J < Hm" Üirl V E[f/2 < E , < f, E2]

71

< lim lim e2 Y" P[i7 , > f/2] = 0.

These last two strings of inequalities prove (2) for z = 3 and thus the lemma.

Q.E.D.

Lemma 3. // \r bAx) = 1 if x £ [a, b) and = Oif x i [a, b), then

(1) lim lim sup E ¿ KkX[0t()ÍUj - X2feX[o>£)(Xnfc)| = 0,fiO «-.CO ¿=1

(2) lim E £ E2^ }(En,) = a2(T)/277-00 ._, I

and

(3) li^ÍE2{UnkX[auJ] = ̂ L\

Proof. In order to prove (1) we observe that

E ZKkX[0jUnk)-X2nkX[oJXttk)\

<£,E{[0<Unk<e]AÍ0<Xnk«],U2nk]

^2¿£Í0^Xnk<í,|En,Vj}+¿E{0<XBk<f,V2fc}.fe=l *=l

Now let 72 —» oo and then í J, 0. The first term on the right-hand side becomes 0

by Lemma 2. The third term on the right-hand side becomes zero by Lemma 1.

Applications of the Cauchy-Schwarz inequality bound the middle term of the

right-hand side by

2 Z^iO^X^f.E^E^X^f,^,

*=l

< 2 | £ F{0 < Xnk < f, E2fc!¡ * j £ filo < Xní < e, V2J |*.



The first of these two multiplied terms is bounded above by oiT) < oo, and the

second tends to zero by Lemma 1 if we first let tz —» oo and then e I 0. This

proves (1). We next prove (2). Since U , is Gaussian with mean zero and vari-

ance a ,, we obtainnk.

*=1 k=l v2*

Now for tz sufficiently large,

b=lV2*

a ,e nK < a3,,nk nk'

from which we obtain that the limit of the first term on the right, as n —» oot is

zero. The limit of the second term, as n —♦ «, is seen to be a (7)/2 by an easy

application of the dominated convergence theorem, and thus (2) is proved. In

order to establish (3), we observe that

Ü» £ *210 < "„k < « " J-J* A" è °2Á^nkte-t2/2dt)2>k=l k=l

and this is seen to equal o ÍT)/2n by an application of the dominated converg-

ence theorem. Q.E.D.

For the convenience of the reader we state here a lemma due to P. Greenwood

and B. Fristedt (Lemma 3.4 in [4]) which we shall use.

Lemma 4. For n = 1, 2," •, let Sn = 2£ = 1 Snk and T' = 2™_j 7nk be sums

of nonnegative independent random variables. Suppose that as n —» oo,

B B

Z u*.*)2 Y,EÍSlk)-.m2<~,k=l k=l

and

Then Var S

lim sup E Z Kk - T2nk\ < 8.

ttz2 - mx,

lim sup Z^)2-^k=l

< [125(772, + «5)]H, lim sup Z^„*>- <s,

and

lim sup |Var 7 - im2 - mx)\ <8+ 12S(ttz, + 8).

We now prove Theorem 4. In the statement of Lemma 4 above, teplace Snfe

^ UnkX[0,e)^„k) and Tnk by Xnk^[0.e){Xnk)' NoW den0te



S(f) = lim sup fi Z KkX[0,()(Unk) - X2nkX[0,e)O<nk)\.k=\

By (1) in Lemma 3, lim,_,0 S(f) = 0. By Lemma 3, 7722 and 772 j in Lemma 4 be-

come 772 2 = a (T)/2 and 772 j = a ÍT)/2tí. Hence by Lemma 4, we have

lim lim supe|o 77-.00

Var(¿XntX[0(f)(X^-I^l^(T) 0,

which proves Theorem 4. Q.E.D.

An immediate consequence of this result is that there always exists a se-

quence of centering constants {el such that ^y_j 1^ -I + cn converges in law.

5. Acknowledgement. We are most grateful to the referee for a detailed

analysis of this paper and suggestions for improvement. The referee's report

provided us with several ways for shortening the paper and led us to obtain a

much better Theorem 1.

REFERENCES

1. S. M. Berman, Sign-invariant random variables and stochastic processes with sign-

invariant increments, Trans. Amer. Math. Soc. 119 (1965), 216-243. MR32 #3113.

2. R. M. Blumenthal and R. K. Getoor, Sample functions of stochastic processes with

stationary independent increments, J. Math. Mech. 10 (1961), 493-516. MR 23 #A689.

3. R. Cogburn and H. G, Tucker, A\ limit theorem for a function of the increments of a

decomposable process, Trans. Amer. Math. Soc. 99 (1961), 278-284. MR 23 #A681.

4. P. Greenwood and B. Fristedt, Variations of processes with stationary independent

increments, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 23 (1972), 171—186.

5. M. Loeve, A l'intérieur du problème central, Publ. Inst. Staust. Univ. Paris

(homage à M. Paul Le'vy) 6 (1957), 313-325. MR 20 #7336.

6. -, Probability theory. Foundations. Random sequences, University Series in

Higher Math., Van Nostrand, Princeton, N. J., 1963. MR 34 #3596.

7. P. W. Millar, Path behavior of processes with stationary independent increments,

Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 17 (1971), 53—73.

8. E. L. Rvaceva, On domains of attraction of multi-dimensional distributions, L'vov.

Gos. Univ. Uc. Zap. 29, Ser. Meh.-Mat. No. 6 (1954), 5-44; English transi., Selected

Transi. Math Statist, and Probability, vol. 2, Amer. Math. Soc, Providence, R. I., 1962,

pp. 183-205. MR 17, 864; 27 #782.

9. H. G. Tucker, O77, asymptotic independence of the partial sums of the positive

parts and negative parts of independent random variables, Advances in Appl. Probability

3 (1971), 404-425. MR 44 #6017.

10. , A graduate course in probability, Probability and Math. Statist., vol. 2,

Academic Press, New York and London, 1967. MR 36 #4593.



11. S. J. Wolfe, Oti moments of infinitely divisible distributions, Ann. Math. Statist.

42 (1971), 2036-2043.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, SANTA BARBARA,CALIFORNIA 93106

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, IRVINE, CALIFORNIA92664 (Current address of H. G. Tucker)

Current address (W. N. Hudson): Department of Mathematics, University of

Utah, Salt Lake City, Utah 84112


LIMIT THEOREMS FOR VARIATIONAL SUMS · tion of all such bivariate limit distributions is obtained....

Documents

Transcript of LIMIT THEOREMS FOR VARIATIONAL SUMS · tion of all such bivariate limit distributions is obtained....